"floating point arithmetic" is a terminology that refer to the arithmetic perform over (finite) representation of real number. See the wikipedia <a href="https://en.wikipedia.org/wiki/Floating-point_arithmetic">article</a> for more details. 

In the <a href="https://en.wikipedia.org/wiki/IEEE_754">formal specification </a> of floating point arithmetic (that should be used by all the major programming languages), it is specified that a "Not A number" (NaN) value should be a number.

If we abstract the fact that floating point arithmetic care only of finitely many values and abstract a bit, we get a sort of arithmetic arithmetic over $\mathbb{R}\cup\{\textrm{NaN}\}$ with $\textrm{NaN}$ being a zero for any arithmetic operation. Formally, for all $x$,  
$$x\times \textrm{NaN} = x + \textrm{NaN} = \frac{x}{0} = \frac{0}{0}= \cdots =\textrm{NaN}$$

Is there any algebraic structures $(E,+,\times)$ having axioms allowing this or is it just to arbitrary to have been introduced even in weird part of algebra?